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Spatio-temporal Lie-Poisson discretization for incompressible magnetohydrodynamics on the sphere
Klas Modin, Michael Roop
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We give a structure preserving spatio-temporal discretization for incompressible magnetohydrodynamics (MHD) on the sphere. Discretization in space is based on the theory of geometric quantization, which yields a spatially discretized analogue of the MHD equations as a finite-dimensional Lie--Poisson system on the dual of the magnetic extension Lie algebra 𝔣=𝔰𝔲(N)⋉𝔰𝔲(N)^*. We also give accompanying structure preserving time discretizations for Lie--Poisson systems on the dual of semi-direct product Lie algebras of the form 𝔣=𝔤⋉𝔤^*, where 𝔤 is a J-quadratic Lie algebra. The time integration method is free of computationally costly matrix exponentials. We prove that the full method preserves a modified Lie--Poisson structure and corresponding Casimir functions, and that the modified structure and Casimirs converge to the continuous ones. The method is demonstrated for two models of magnetic fluids: incompressible magnetohydrodynamics and Hazeltine's model.
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